Okay, I google'd both flexigon and flexagon, too, and I saw lots of things based on triangles, but I didn't find much of anything based on squares, and what I did wasn't this.
Here's the single image that I was looking at on the site above:
And here's how I figured it out to work. Looking at this image for an example, crease a piece of paper so there's a grid of four squares x four squares. (If you use an 8.5x11, you get rectangles instead of squares, but that doesn't matter ... the point is, you've got 16 boxes of uniform shape in four rows of four.) For my explanation purposes, label each box with a number, starting at the top left, 1,2,3,4 in the first row, 5,6,7,8 in the second, continuing until you get to the bottom row, which will be 13,14,15,16. Okay? (If that isn't clear, looking at this picture of Patti's, the palmistry stamp is box 1, the romantic pic of the girl & guy is box 2, the tree woman is box 3, the square in the corner with the book cover attached is box 4, and the "I see passion" is box 5. Got it?)
Use a scissors to start a squared-off spiral shape cut, starting at the bottom of box 4 (between box 4 and 8) and cutting straight across that line, stopping at the intersection where you reach box 1 and 5. Then, start cutting straight downward, between box 5 and 6 (Patti's "I see passion" and "Sweet Dreams"), then between 9 and 10. At the bottom of box 10, stop cutting downwards and start cutting to the right again, between box 10 and 14, continuing between 11 and 15, then turning up again, cutting between 11 and 12, then turning AGAIN and cutting along the top of box 11, between 11 and 7. When you look at it, you've kinda cut a squared-off spiral, right?
Then,, starting at one end or the other, simply start folding between the squares/boxes to make sort of an accordian fold, back and forth, or up and down when you get to the appropriate points (this will make sense when you actually start folding).
Patti's example as shown in the link in this post is actually the back side, I figured out after I posted last night. The front isn't shown in full, but several of the painted boxes/spreads shown on that page I originally linked are from the "right side" of this piece. (You can see a bit of the "antique map" book covers in the backgrounds of some of the pix.)
I decided, after playing with it awhile late last night, that it's a little wierd, but also fun for images, kinda collage-y, kinda altered-book-ish. Doesn't really make sense for text, partly because of the twists and turns. (I could be proved otherwise.)
And I still think it would be a very fun Glitter swap, if anyone's interested.
And now my boys are demanding my attention, so I gotta go. Oh yeah, the "flutter book" idea, as I understand it, defines anything folded accordian style so that on a windy day, all the pages could flutter out of the book in the wind. *shrug*